Modular links: bunch algorithm and upper volume bounds

Hui, Connie On Yu; Rodríguez Migueles, José Andrés

doi:10.2140/agt.2026.26.2123

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Modular links: bunch algorithm and upper volume bounds

Connie On Yu Hui and José Andrés Rodríguez Migueles

Algebraic & Geometric Topology 26 (2026) 2123–2156

DOI: 10.2140/agt.2026.26.2123

Abstract

In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce an improved algorithm, which we call the bunch algorithm, to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

Keywords

hyperbolic volume, bunch algorithm, modular links, Lorenz links, knot theory, template, William's algorithm, braid index, code word, word period

Mathematical Subject Classification

Primary: 57K10, 57K32

Secondary: 37D40, 37E35

References

Publication

Received: 29 February 2024

Revised: 27 April 2025

Accepted: 29 May 2025

Published: 22 June 2026

Authors

Connie On Yu Hui
School of Mathematics Monash University Melbourne Australia

José Andrés Rodríguez Migueles
Centro de Investigación en Matemáticas Guanajuato Mexico

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Volume 26, issue 6 (2026)